Complex Harmonics and the Spherical Tensor Gradient Operator

1 Complex Harmonics and the Spherical Tensor Gradient Operator 

The complex spherical harmonic y/m(f2) is related to the associated Legendre polynomial Pim[x] by 

The spherical tensor gradient operator (STGO) is a solid harmonic whose Cartesian coordinate arguments have been replaced by Cartesian derivatives. Hobson s theorem [25] is the result of acting a STGO upon any spherical function /(r ) 

And when acted upon another solid harmonic, one obtains the following STGO differentiation rules [4, 47] 

The utility of the above rules are numerous however, the reader may gain a better appreciation upon considering two brief examples. We can express the translation of a regular or irregular solid harmonic with a Taylor series expansion. 

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